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  • A tangent is drawn to the ellipse <img alt="\mathrm{\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cfrac%7Bx%5E%7B2%7D%7D%7Ba%5E%7B2%

A tangent is drawn to the ellipse \mathrm{\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1} at point \mathrm{P} uch that it intersect the auxiliary circle at points A and B, if \mathrm{S} and \mathrm{S^{\prime}}are the focii of the ellipse, then the minimum value of \mathrm{S^{\prime} A+S^{\prime} B} is

Option: 1

b


Option: 2

2b


Option: 3

2a


Option: 4

none of these


Answers (1)

best_answer

We know S'A.S'B \mathrm{=b^{2}}

Applying \mathrm{\mathrm{AM} \geq \mathrm{GM}}
\mathrm{\frac{S^{\prime} A+S^{\prime} B}{2} \geq \sqrt{S^{\prime} A . S^{\prime} B}}

Clearly \mathrm{S'A + S'B\geq 2 b}

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